In Situ Ultra-Small- and Small-Angle X-ray Scattering Study of ZnO Nanoparticle Formation and Growth through Chemical Bath Deposition in the Presence of Polyvinylpyrrolidone

ZnO inverse opals combine the outstanding properties of the semiconductor ZnO with the high surface area of the open-porous framework, making them valuable photonic and catalysis support materials. One route to produce inverse opals is to mineralize the voids of close-packed polymer nanoparticle templates by chemical bath deposition (CBD) using a ZnO precursor solution, followed by template removal. To ensure synthesis control, the formation and growth of ZnO nanoparticles in a precursor solution containing the organic additive polyvinylpyrrolidone (PVP) was investigated by in situ ultra-small- and small-angle X-ray scattering (USAXS/SAXS). Before that, we studied the precursor solution by in-house SAXS at T = 25 °C, revealing the presence of a PVP network with semiflexible chain behavior. Heating the precursor solution to 58 °C or 63 °C initiates the formation of small ZnO nanoparticles that cluster together, as shown by complementary transmission electron microscopy images (TEM) taken after synthesis. The underlying kinetics of this process could be deciphered by quantitatively analyzing the USAXS/SAXS data considering the scattering contributions of particles, clusters, and the PVP network. A nearly quantitative description of both the nucleation and growth period could be achieved using the two-step Finke–Watzky model with slow, continuous nucleation followed by autocatalytic growth.


Introduction
The field of artificial, structured organic-inorganic hybrid materials has received enormous attention in the last few decades given the major contributions in numerous fields such as coatings, catalysis, sensors, and electronics [1][2][3]. In terms of fabrication, one straightforward approach is the use of organic templates, including carbon nanotubes [4,5], polymers [6,7], or even biological objects such as viruses [8,9], on which an inorganic matrix is deposited in situ. To preserve the delicate template structure, mild deposition conditions are necessary. Next to the frequently used sol-gel procedure [6,10,11], chemical bath deposition (CBD) has been proven beneficial, in particular for the synthesis of nanostructured hybrid materials with a finely tuned thickness of the deposited inorganic phase on the organic template. In CBD, an (organic) template is mineralized while being immersed in a precursor solution at moderate temperatures, typically below 100 • C [12]. Additives such as polymers [13] or small molecules like amino acids [14] may be present and used as a structure-directing agent to precisely control the growth of the solid with respect to size, morphology, and crystallinity. Such control is achieved by the ability of the additives to stabilize the dissolved ions, direct and restrict the particle growth, and prevent nanoparticle agglomeration by steric and/or electrostatic interactions. Using this approach, dense and smooth thin films and nanostructured hybrid materials of metal chalcogenides like CdS [15][16][17], TiO 2 [18][19][20], and ZnO [21][22][23] were fabricated through organic templates. Among them, the transparent semiconductor ZnO is of particular interest as it covers a vast range of applications from optoelectronics [24,25] to gas sensors [26,27] and lasing devices [28,29], etc.
Applying a well-established CBD route and using various organic templates such as self-assembled monolayers [30][31][32][33], polymers [32,34], polymer foams [35], DNA [36], and viruses [37][38][39][40], ZnO hybrid materials were prepared from a methanolic ZnO precursor solution in the presence of polyvinylpyrrolidone (PVP). In addition, close-packed assemblies of polymer nanoparticles were mineralized by applying the same approach, followed by the removal of the particles to produce ZnO inverse opals [41,42]. They are a promising candidate for photonic applications [43,44] or as catalyst solid supports in heterogeneous catalysis [45], combining the ordered porous structure with the favorable electrical and optical properties of ZnO. First insights into the role of PVP on the deposition rate, mechanism, and film morphology of ZnO films obtained applying this CBD approach were given in the studies of Lipowsky et al. [30,31,33]. It was found that in the absence of PVP, the immediate precipitation of micron-sized ZnO particles takes place, whereas intermediate PVP volume fractions (0.06 < φ PVP < 0.14) are required for the successful preparation of thin, homogeneous nanocrystalline ZnO films. It was also shown that the deposition of ZnO occurs via a transient amorphous phase in which needle-like particles containing both zinc and PVP are formed [33], similar to the well-known formation of intermediate amorphous phases in biomineralization and bioinspired mineralization processes [46,47].
A powerful technique for the in situ study of the structural evolution during nanoparticle synthesis is time-resolved small-angle X-ray scattering (SAXS), which can be complemented by ex situ imaging techniques such as transmission electron microscopy (TEM). As far as ZnO formation via the sol-gel method is concerned, Caetano et al. investigated the nucleation and growth of ZnO nanoparticles synthesized through an additive-free sol-gel process via time-resolved tandem SAXS/UV-Vis and X-ray adsorption fine structure (XAFS) experiments [48][49][50]. These studies revealed that ZnO nuclei formed by supersaturationinduced nucleation are subject to the so-called oriented attachment growth [51,52], where particles aggregate by coalescence. In the late stages, Ostwald ripening [53,54] with dissolution and reprecipitation processes dominates. Analogies were found by Herbst et al. during the thermolysis of Zn-oleate precursors, verified by in situ SAXS/WAXS/UV-Vis measurements [55]. In their study, a classical homogeneous nucleation and growth model was applied to describe the process of ZnO nanoparticle synthesis. A nonclassical growth behavior where both diffusion-and surface-control occur was observed when the PVP-mediated solvothermal synthesis of ZnO nanorods was studied by ex situ SAXS and TEM [56].
In this study, we use time-resolved USAXS and SAXS for the in situ investigation of the role of the organic additive PVP during the formation and growth of individual ZnO nanoparticles and ZnO clusters in a methanolic precursor solution applying the abovementioned CBD method. As found in previous studies, this method requires a relatively high volume fraction of PVP (φ PVP = 7.1 × 10 −2 ); therefore, its scattering contribution must be considered when analyzing the scattering data. Accordingly, the first part of this work deals with the study of the polymer structure in the precursor solution before initiating the synthesis of ZnO particles by heating. In the second part, we clarified the ZnO particle formation performed at two temperatures, i.e., 58 • C and 63 • C, via time-resolved synchrotron USAXS measurements. Form factor modeling and modelindependent invariant analyses of the recorded USAXS data allowed us to not only describe the data quantitatively but also to unravel the ZnO nanoparticle formation mechanism.

Synthesis of ZnO Nanoparticles
ZnO nanoparticles were synthesized following an established procedure [35,41,42]. Briefly, three methanolic stock solutions of zinc acetate dihydrate (ZnAc 2 × 2 H 2 O) (37.4 mM, Sigma-Aldrich, St. Louis, MO, USA), tetraethylammonium hydroxide (TEAOH) (79.0 mM, Sigma-Aldrich), and polyvinylpyrrolidone (PVP) (25.7 mM) (Sigma-Aldrich, M w = 10,000 g/mol (10 k), Lot # BCBJ4889V) were used to prepare the precursor solution. While the stock solutions can be stored for a longer period of time, the precursor solution was prepared freshly before use to avoid unwanted ZnO formation. Equal volumes of the ZnAc 2 and PVP stock solution were mixed, followed by dropwise addition ( For the synthesis of ZnO particles, the precursor solution was heated to T = 58 • C and 63 • C, respectively.

In-House Small-Angle X-ray Scattering
Room temperature studies of the precursor solution were performed on a SAXSess instrument from Anton Paar (Graz, Austria). The instrument was equipped with a Cu K α X-ray source (λ = 1.54 Å) operated in line collimation mode. Samples were filled into a quartz capillary flow cell and data were collected at 25 • C using imaging plates from Fujifilm (Greenwood, SC, USA), which were read by a cyclone scanner from PerkinElmer (Covina, CA, USA). One-dimensional corrected and reduced scattering profiles I(q) as a function of the absolute value of the scattering vector q, with q = 4π/λ sin(θ/2), where θ is the scattering angle, were obtained using Anton Paar's SAXSquant 2D and 1D software, including the subtraction of the dark current and the empty cell scattering. To normalize the data to an absolute scale, water was used as a secondary calibration standard [57]. The scattering data were analyzed using SasView version 5.0.4, with slit-smearing accounted for in the fit model. Alternatively, the slit-smeared data were desmeared using SAXSquant 2.0.

Ultra-Small-and Small-Angle Synchrotron X-ray Scattering
USAXS and SAXS data were collected at the 9-ID-C beamline at the Advanced Photon Source at Argonne National Laboratory (Argonne, IL, USA) [58,59]. A combined q range between 3.7 × 10 −4 Å −1 and 1.7 Å −1 was covered using an X-ray energy of 21 keV (λ = 0.5895 Å) and an X-ray flux of about 5 × 10 12 photons mm −2 s −1 . To study the time-dependent growth of the ZnO particles, the precursor solution was filled into an NMR tube (Wilmad-LabGlass TM , WG-1000-4, inner diameter 4 mm), which was placed in a thermostated home-built cell holder. USAXS scattering data were recorded starting at T = 25 • C, followed by rapid heating (1 K/s) to T = 58 • C and 63 • C, respectively, with collection times of 100 s each. Before and after the time-resolved USAXS measurements, SAXS patterns were acquired with collection times of 20 s.
The slit-smeared USAXS data were reduced for instrumental background and empty cell scattering, normalized to absolute scale, and desmeared using the instrument data reduction software package Indra for Igor Pro 9.0. For the SAXS data and to combine the USAXS and SAXS data, Nika [60] and Irena [61] were used, respectively. The scattering data were analyzed using SasView version 5.0.4, with the slit-smearing accounted for in the fit model.

Transmission Electron Microscopy
For particle visualization, TEM images were collected on a EM10 from Zeiss (Jena, Germany) operated at 60 kV. Each nanoparticle solution was diluted in methanol at room temperature. A small drop of the diluted sample was placed on a TEM copper grid covered with a thin holey carbon film. The drop was then gently blotted to remove the excess In the following, we first describe the structural investigation of PVP in the methanolic ZnO precursor solution by in-house SAXS and USAXS/SAXS. We then discuss the timeresolved synchrotron USAXS measurements performed to determine the kinetic processes of ZnO particle formation and growth in situ. We show that a quantitative analysis of all scattering data is possible when considering the scattering contributions of the previously determined stabilizing polymer structure and two ZnO species, i.e., ZnO nanoparticles and their clusters. In the last part, we use the determined time evolution of the size and volume fraction of ZnO particle clusters to study the kinetics using the two-step Finke-Watzky model [62][63][64][65].

PVP Structure in the Precursor Solution
The chemical bath deposition (CBD) of ZnO nanoparticles studied in this work is based on the thermally induced hydrolysis of the zinc precursor zinc acetate (ZnAc 2 ) in a methanolic solution under basic conditions adjusted by the organic base tetraethylammonium hydroxide (TEAOH). Furthermore, the ion-stabilizing and structure-directing agent PVP (10 k) is added at a volume fraction of φ PVP = 7.1 × 10 −2 . As a prerequisite for the comprehensive investigation of the ZnO particle formation by USAXS/SAXS, the contribution of all components to the total scattering should be clarified first. Therefore, we started with the investigation of the PVP structure in methanol before and after the addition of the respective components as well as the PVP structure in the final composition of the precursor solution by SAXS at 25 • C.
In Figure 1a, the recorded slit-smeared SAXS intensities I(q) were plotted as a function of the scattering vector q using a double-logarithmic representation. The scattering curve of PVP in pure methanol φ PVP = 7.1 × 10 −2 (8.6 mM) ( In Figure 1a, the recorded slit-smeared SAXS intensities I(q) were plotted as a function of the scattering vector q using a double-logarithmic representation. The scattering curve of PVP in pure methanol ϕPVP = 7.1 × 10 −2 (8.6 mM) ( ) shows a Guinier region at low q with almost constant scattering intensity. As q increases, a power law decay is observed until a minimum is reached at q ≈ 1 Å -1 , followed by the intermolecular C-C chain interaction peak [66] at q ≈ 1.5 Å −1 . Similar behavior was found when 12.5 mM ZnAc2 ( ) or 26.3 mM TEAOH ( )were added while keeping the polymer amount constant at ϕPVP = 7.1 × 10 −2 . However, for the final composition of the precursor solution ( ), instead of a Guinier region for q < 0.1 Å −1 , a power law decay with I(q) ≈ q −1.4 can be seen. Thus, the presence of both ZnAc2 and TEAOH results in a significantly different structure of the polymer.
(where both ZnAc2 and TEAOH are present) in more detail, the power law decay (I(q) ≈ q −1.4 ) recorded in the low q limit of the in-house SAXS instrument provides little information on the global structure and size of the polymer. This information can be obtained by extending the q-range towards much smaller values using the USAXS/SAXS setup at the 9-ID-C beamline at the Advanced Photon Source at the Argonne National Laboratory [58,59].
The inset in Figure 1 shows a good agreement between the desmeared in-house ( ) and USAXS/SAXS ( ) data. In addition, the combination of the data clearly shows the transition of the power law into a Guinier region at q < 0.01 Å −1 . Utilizing Guinier's approximation for low q (ln I(q) ≈ ln I0 − (Rg 2 /3) q 2 ), we obtained Rg(PVP) = 6.3 ± 0.3 nm, which corresponds to a fourfold increase compared to Rg(PVP) in pure methanol and in solutions containing only ZnAc2 or TEAOH. This significant change in the PVP structure caused by the increased ion concentration due to the presence of both ZnO precursor and base is most likely related to a stronger repulsion of the segments within the polymer chain, leading to a further increase in the chain stiffness. This hypothesis is supported by the increase in mesh size determined from the analysis at medium q's to ξPVP = 2.9 ± 0.2 nm. Moreover, ZnAc2 might start to hydrolyze even at 25 °C, changing its chain flexibility so that the first particle nuclei could have formed within the PVP. However, this reaction is significantly accelerated only by increasing the temperature. ) shows a Guinier region at low q with almost constant scattering intensity. As q increases, a power law decay is observed until a minimum is reached at q ≈ 1 Å −1 , followed by the intermolecular C-C chain interaction peak [66] at q ≈ 1.5 Å −1 . Similar behavior was found when 12.5 mM ZnAc 2 ( In Figure 1a, the recorded slit-smeared SAXS intensities I(q) were plotted as a function of the scattering vector q using a double-logarithmic representation. The scattering curve of PVP in pure methanol ϕPVP = 7.1 × 10 −2 (8.6 mM) ( ) shows a Guinier region at low q with almost constant scattering intensity. As q increases, a power law decay is observed until a minimum is reached at q ≈ 1 Å -1 , followed by the intermolecular C-C chain interaction peak [66] at q ≈ 1.5 Å −1 . Similar behavior was found when 12.5 mM ZnAc2 ( ) or 26.3 mM TEAOH ( )were added while keeping the polymer amount constant at ϕPVP = 7.1 × 10 −2 . However, for the final composition of the precursor solution ( ), instead of a Guinier region for q < 0.1 Å −1 , a power law decay with I(q) ≈ q −1.4 can be seen. Thus, the presence of both ZnAc2 and TEAOH results in a significantly different structure of the polymer.
(where both ZnAc2 and TEAOH are present) in more detail, the power law decay (I(q) ≈ q −1.4 ) recorded in the low q limit of the in-house SAXS instrument provides little information on the global structure and size of the polymer. This information can be obtained by extending the q-range towards much smaller values using the USAXS/SAXS setup at the 9-ID-C beamline at the Advanced Photon Source at the Argonne National Laboratory [58,59]. The inset in Figure 1 shows a good agreement between the desmeared in-house ( ) and USAXS/SAXS ( ) data. In addition, the combination of the data clearly shows the transition of the power law into a Guinier region at q < 0.01 Å −1 . Utilizing Guinier's approximation for low q (ln I(q) ≈ ln I0 − (Rg 2 /3) q 2 ), we obtained Rg(PVP) = 6.3 ± 0.3 nm, which corresponds to a fourfold increase compared to Rg(PVP) in pure methanol and in solutions containing only ZnAc2 or TEAOH. This significant change in the PVP structure caused by the increased ion concentration due to the presence of both ZnO precursor and base is most likely related to a stronger repulsion of the segments within the polymer chain, leading to a further increase in the chain stiffness. This hypothesis is supported by the increase in mesh size determined from the analysis at medium q's to ξPVP = 2.9 ± 0.2 nm. Moreover, ZnAc2 might start to hydrolyze even at 25 °C, changing its chain flexibility so that the first particle nuclei could have formed within the PVP. However, this reaction is significantly accelerated only by increasing the temperature. In Figure 1a, the recorded slit-smeared SAXS intensities I(q) were plotted as a function of the scattering vector q using a double-logarithmic representation. The scattering curve of PVP in pure methanol ϕPVP = 7.1 × 10 −2 (8.6 mM) ( ) shows a Guinier region at low q with almost constant scattering intensity. As q increases, a power law decay is observed until a minimum is reached at q ≈ 1 Å -1 , followed by the intermolecular C-C chain interaction peak [66] at q ≈ 1.5 Å −1 . Similar behavior was found when 12.5 mM ZnAc2 ( ) or 26.3 mM TEAOH ( )were added while keeping the polymer amount constant at ϕPVP = 7.1 × 10 −2 . However, for the final composition of the precursor solution ( ), instead of a Guinier region for q < 0.1 Å −1 , a power law decay with I(q) ≈ q −1.4 can be seen. Thus, the presence of both ZnAc2 and TEAOH results in a significantly different structure of the polymer.
(where both ZnAc2 and TEAOH are present) in more detail, the power law decay (I(q) ≈ q −1.4 ) recorded in the low q limit of the in-house SAXS instrument provides little information on the global structure and size of the polymer. This information can be obtained by extending the q-range towards much smaller values using the USAXS/SAXS setup at the 9-ID-C beamline at the Advanced Photon Source at the Argonne National Laboratory [58,59].
The inset in Figure 1 shows a good agreement between the desmeared in-house ( ) and USAXS/SAXS ( ) data. In addition, the combination of the data clearly shows the transition of the power law into a Guinier region at q < 0.01 Å −1 . Utilizing Guinier's approximation for low q (ln I(q) ≈ ln I0 − (Rg 2 /3) q 2 ), we obtained Rg(PVP) = 6.3 ± 0.3 nm, which corresponds to a fourfold increase compared to Rg(PVP) in pure methanol and in solutions containing only ZnAc2 or TEAOH. This significant change in the PVP structure caused by the increased ion concentration due to the presence of both ZnO precursor and base is most likely related to a stronger repulsion of the segments within the polymer chain, leading to a further increase in the chain stiffness. This hypothesis is supported by the increase in mesh size determined from the analysis at medium q's to ξPVP = 2.9 ± 0.2 nm. Moreover, ZnAc2 might start to hydrolyze even at 25 °C, changing its chain flexibility so that the first particle nuclei could have formed within the PVP. However, this reaction is significantly accelerated only by increasing the temperature. In Figure 1a, the recorded slit-smeared SAXS intensities I(q) were plotted as a function of the scattering vector q using a double-logarithmic representation. The scattering curve of PVP in pure methanol ϕPVP = 7.1 × 10 −2 (8.6 mM) ( ) shows a Guinier region at low q with almost constant scattering intensity. As q increases, a power law decay is observed until a minimum is reached at q ≈ 1 Å -1 , followed by the intermolecular C-C chain interaction peak [66] at q ≈ 1.5 Å −1 . Similar behavior was found when 12.5 mM ZnAc2 ( ) or 26.3 mM TEAOH ( )were added while keeping the polymer amount constant at ϕPVP = 7.1 × 10 −2 . However, for the final composition of the precursor solution ( ), instead of a Guinier region for q < 0.1 Å −1 , a power law decay with I(q) ≈ q −1.4 can be seen. Thus, the presence of both ZnAc2 and TEAOH results in a significantly different structure of the polymer.
(where both ZnAc2 and TEAOH are present) in more detail, the power law decay (I(q) ≈ q −1.4 ) recorded in the low q limit of the in-house SAXS instrument provides little information on the global structure and size of the polymer. This information can be obtained by extending the q-range towards much smaller values using the USAXS/SAXS setup at the 9-ID-C beamline at the Advanced Photon Source at the Argonne National Laboratory [58,59].
The inset in Figure 1 shows a good agreement between the desmeared in-house ( ) and USAXS/SAXS ( ) data. In addition, the combination of the data clearly shows the transition of the power law into a Guinier region at q < 0.01 Å −1 . Utilizing Guinier's approximation for low q (ln I(q) ≈ ln I0 − (Rg 2 /3) q 2 ), we obtained Rg(PVP) = 6.3 ± 0.3 nm, which corresponds to a fourfold increase compared to Rg(PVP) in pure methanol and in solutions containing only ZnAc2 or TEAOH. This significant change in the PVP structure caused by the increased ion concentration due to the presence of both ZnO precursor and base is most likely related to a stronger repulsion of the segments within the polymer chain, leading to a further increase in the chain stiffness. This hypothesis is supported by the increase in mesh size determined from the analysis at medium q's to ξPVP = 2.9 ± 0.2 nm. Moreover, ZnAc2 might start to hydrolyze even at 25 °C, changing its chain flexibility so that the first particle nuclei could have formed within the PVP. However, this reaction is significantly accelerated only by increasing the temperature. ), instead of a Guinier region for q < 0.1 Å −1 , a power law decay with I(q) ≈ q −1.4 can be seen. Thus, the presence of both ZnAc 2 and TEAOH results in a significantly different structure of the polymer.
In general, the recorded scattering curves provide information about the polymer structure on different-length scales: the Guinier region gives the overall dimension of the polymer, i.e., its radius of gyration R g (PVP), while the exponent m of the power law decay is related to the polymer-solvent interactions [67,68]. Furthermore, the correlation length ξ, quantifying the end-to-end distance of a coil or the mesh size, i.e., the mean diameter of a cavity within the polymer network [69], respectively, can be extracted.
To determine R g (PVP) and the exponent m of the power law decay, the scattering curves, except for the final composition, were analyzed according to the polymers with excluded volume effects model [67,68]: where I 0 corresponds to the forward scattering intensity, γ is the incomplete gamma function, and the parameter U is defined by Since I(q) poly does not account for large q features such as the intermolecular C-C chain interaction peak [66], Equation (1) was combined with a Gaussian I(q) G , which finally leads to the following expression: Herein, G is the scaling factor for the Gaussian with the maximum peak position at q G , and the peak width σ G . Note that the solid lines in Figure 1a  In Figure 1a, the recorded slit-smeared SAXS intensities I(q) were plotted as a function he scattering vector q using a double-logarithmic representation. The scattering curve of in pure methanol ϕPVP = 7.1 × 10 −2 (8.6 mM) ( ) shows a Guinier region at low q h almost constant scattering intensity. As q increases, a power law decay is observed until inimum is reached at q ≈ 1 Å -1 , followed by the intermolecular C-C chain interaction peak at q ≈ 1.5 Å −1 . Similar behavior was found when 12.5 mM ZnAc2 ( ) or 26.3 TEAOH ( )were added while keeping the polymer amount constant at ϕPVP 1 × 10 −2 . However, for the final composition of the precursor solution ( ), ead of a Guinier region for q < 0.1 Å −1 , a power law decay with I(q) ≈ q −1.4 can be seen. s, the presence of both ZnAc2 and TEAOH results in a significantly different structure he polymer.
(where both ZnAc2 and TEAOH are present) in more detail, the power law decay (I(q) 1.4 ) recorded in the low q limit of the in-house SAXS instrument provides little inforion on the global structure and size of the polymer. This information can be obtained by nding the q-range towards much smaller values using the USAXS/SAXS setup at the 9-C beamline at the Advanced Photon Source at the Argonne National Laboratory [58,59]. inset in Figure 1 shows a good agreement between the desmeared in-house ( d USAXS/SAXS ( ) data. In addition, the combination of the data rly shows the transition of the power law into a Guinier region at q < 0.01 Å −1 . Utilizing inier's approximation for low q (ln I(q) ≈ ln I0 − (Rg 2 /3) q 2 ), we obtained Rg(PVP) = 6.3 ± nm, which corresponds to a fourfold increase compared to Rg(PVP) in pure methanol in solutions containing only ZnAc2 or TEAOH. This significant change in the PVP cture caused by the increased ion concentration due to the presence of both ZnO presor and base is most likely related to a stronger repulsion of the segments within the ymer chain, leading to a further increase in the chain stiffness. This hypothesis is supted by the increase in mesh size determined from the analysis at medium q's to ξPVP = ± 0.2 nm. Moreover, ZnAc2 might start to hydrolyze even at 25 °C, changing its chain ibility so that the first particle nuclei could have formed within the PVP. However, this ction is significantly accelerated only by increasing the temperature. The FW model assumes continuous nucleation of ZnO particles. This can be indeed erved in the volume fraction profile of the ZnO particles ( ) for T = 58 °C in Figure 6, ere an initial increase is followed by reaching an almost constant value, while the vole fraction of the cluster ( ) follows a sigmoidal profile. These results suggest that new particles are formed continuously throughout the reaction, feeding the clusters, ich in turn grow simultaneously. Hence, at intermediate times, an equilibrium is estabed between ZnO particles' formation and consumption. ere ∆ 2 = |ρZnO − ρMeOH | 2 . In Figure 6, the obtained cluster volume fraction ϕZnO-clusv(t) ( ) for T = 58 °C is plotted as a function of time. Comparing the cluster volume ). The scattering curves were described using the polymers with excluded volume effects model [67,68], and for the final composition, the flexible cylinder model [70,71] combined with a Gaussian, respectively (solid lines). To help the reader distinguish between the scattering curves, they are displaced by an arbitrary factor. Inset: Comparison of desmeared in-house SAXS and USAXS/SAXS data for the final composition with the flexible cylinder fit. (b) Applied fit models (Equations (3) and (4)) and their respective scattering contributions are shown for PVP in methanol.
To determine Rg(PVP) and the exponent m of the power law decay, the scattering curves, except for the final composition, were analyzed according to the polymers with excluded volume effects model [67,68]: where I0 corresponds to the forward scattering intensity, is the incomplete gamma function, and the parameter is defined by Since I(q)poly does not account for large q features such as the intermolecular C-C chain interaction peak [66], Equation (1) was combined with a Gaussian I(q)G, which finally leads , measured at T = 25 • C. The scattering curves were described using the polymers with excluded volume effects model [67,68], and for the final composition, the flexible cylinder model [70,71] combined with a Gaussian, respectively (solid lines). To help the reader distinguish between the scattering curves, they are displaced by an arbitrary factor. Inset: Comparison of desmeared in-house SAXS and USAXS/SAXS data for the final composition with the flexible cylinder fit. (b) Applied fit models (Equations (3) and (4)) and their respective scattering contributions are shown for PVP in methanol.
The correlation length ξ was obtained, separately, by applying the correlation length model [69] I(q) ξ , which was again combined with a Gaussian: where the first term accounts for the scattering of large-scale polymer clusters, and the second term for the polymer chains. A and C are scaling parameters and m and b are the exponent of the power law decay( denotes for small q s) and the Lorentzian exponent, respectively. In cases where the low q power law decay was absent, the first term was In Figure 1a, the recorded slit-smeared SAXS intensities I(q) were plotted as a function of the scattering vector q using a double-logarithmic representation. The scattering curve of PVP in pure methanol ϕPVP = 7.1 × 10 −2 (8.6 mM) ( ) shows a Guinier region at low q with almost constant scattering intensity. As q increases, a power law decay is observed until a minimum is reached at q ≈ 1 Å -1 , followed by the intermolecular C-C chain interaction peak [66] at q ≈ 1.5 Å −1 . Similar behavior was found when 12.5 mM ZnAc2 ( ) or 26.3 mM TEAOH ( )were added while keeping the polymer amount constant at ϕPVP = 7.1 × 10 −2 . However, for the final composition of the precursor solution ( ), instead of a Guinier region for q < 0.1 Å −1 , a power law decay with I(q) ≈ q −1.4 can be seen. Thus, the presence of both ZnAc2 and TEAOH results in a significantly different structure of the polymer.
(where both ZnAc2 and TEAOH are present) in more detail, the power law decay (I(q) ≈ q −1.4 ) recorded in the low q limit of the in-house SAXS instrument provides little information on the global structure and size of the polymer. This information can be obtained by extending the q-range towards much smaller values using the USAXS/SAXS setup at the 9-ID-C beamline at the Advanced Photon Source at the Argonne National Laboratory [58,59].
The inset in Figure 1 shows a good agreement between the desmeared in-house ( ) and USAXS/SAXS ( ) data. In addition, the combination of the data clearly shows the transition of the power law into a Guinier region at q < 0.01 Å −1 . Utilizing Guinier's approximation for low q (ln I(q) ≈ ln I0 − (Rg 2 /3) q 2 ), we obtained Rg(PVP) = 6.3 ± 0.3 nm, which corresponds to a fourfold increase compared to Rg(PVP) in pure methanol and in solutions containing only ZnAc2 or TEAOH. This significant change in the PVP structure caused by the increased ion concentration due to the presence of both ZnO precursor and base is most likely related to a stronger repulsion of the segments within the polymer chain, leading to a further increase in the chain stiffness. This hypothesis is supported by the increase in mesh size determined from the analysis at medium q's to ξPVP = 2.9 ± 0.2 nm. Moreover, ZnAc2 might start to hydrolyze even at 25 °C, changing its chain flexibility so that the first particle nuclei could have formed within the PVP. However, this reaction is significantly accelerated only by increasing the temperature. The FW model assumes continuous nucleation of ZnO particles. This can be indeed observed in the volume fraction profile of the ZnO particles ( ) for T = 58 °C in Figure 6, where an initial increase is followed by reaching an almost constant value, while the volume fraction of the cluster ( ) follows a sigmoidal profile. These results suggest that new ZnO particles are formed continuously throughout the reaction, feeding the clusters, which in turn grow simultaneously. Hence, at intermediate times, an equilibrium is established between ZnO particles' formation and consumption. where ∆ 2 = |ρZnO − ρMeOH | 2 . In Figure 6, the obtained cluster volume fraction ϕZnO-cluster,inv(t) ( ) for T = 58 °C is plotted as a function of time. Comparing the cluster volume fractions obtained from the model analysis of the USAXS data and the invariant, i.e., ϕZnOcluster,I(q) and ϕZnO-cluster,inv, slightly lower values are found for the latter, which might be a consequence of the finite q-range used for the invariant determination as well as the approximation of spherical clusters with compact packing of ZnO particles and a homogeneous scattering length density distribution.
) in Figure 1b. Applying Equations (3) and (4) and taking the slit-smearing into account, it was possible to describe the scattering curves quantitatively. The best-fit parameters are compiled in Tables S1 and S2 in the Supplementary Materials, while the corresponding Guinier radius R g (PVP), the exponent m of the power law decay at high q s, and the correlation length ξ PVP are summarized in Table 1. Following the analysis of the scattering data of PVP in methanol ( In Figure 1a, the recorded slit-smeared SAXS intensities I(q) were plotted as a function of the scattering vector q using a double-logarithmic representation. The scattering curve of PVP in pure methanol ϕPVP = 7.1 × 10 −2 (8.6 mM) ( ) shows a Guinier region at low q with almost constant scattering intensity. As q increases, a power law decay is observed until a minimum is reached at q ≈ 1 Å -1 , followed by the intermolecular C-C chain interaction peak [66] at q ≈ 1.5 Å −1 . Similar behavior was found when 12.5 mM ZnAc2 ( ) or 26.3 mM TEAOH ( )were added while keeping the polymer amount constant at ϕPVP = 7.1 × 10 −2 . However, for the final composition of the precursor solution ( ), instead of a Guinier region for q < 0.1 Å −1 , a power law decay with I(q) ≈ q −1.4 can be seen.
Thus, the presence of both ZnAc2 and TEAOH results in a significantly different structure of the polymer.
(where both ZnAc2 and TEAOH are present) in more detail, the power law decay (I(q) ≈ q −1.4 ) recorded in the low q limit of the in-house SAXS instrument provides little infor-mation on the global structure and size of the polymer. This information can be obtained by extending the q-range towards much smaller values using the USAXS/SAXS setup at the 9-ID-C beamline at the Advanced Photon Source at the Argonne National Laboratory [58,59].
The inset in Figure 1 shows a good agreement between the desmeared in-house ( ) and USAXS/SAXS ( ) data. In addition, the combination of the data clearly shows the transition of the power law into a Guinier region at q < 0.01 Å −1 . Utilizing Guinier's approximation for low q (ln I(q) ≈ ln I0 − (Rg 2 /3) q 2 ), we obtained Rg(PVP) = 6.3 ± 0.3 nm, which corresponds to a fourfold increase compared to Rg(PVP) in pure methanol and in solutions containing only ZnAc2 or TEAOH. This significant change in the PVP structure caused by the increased ion concentration due to the presence of both ZnO pre-cursor and base is most likely related to a stronger repulsion of the segments within the polymer chain, leading to a further increase in the chain stiffness. This hypothesis is sup-ported by the increase in mesh size determined from the analysis at medium q's to ξPVP = 2.9 ± 0.2 nm. Moreover, ZnAc2 might start to hydrolyze even at 25 °C, changing its chain flexibility so that the first particle nuclei could have formed within the PVP. However, this reaction is significantly accelerated only by increasing the temperature. The FW model assumes continuous nucleation of ZnO particles. This can be indeed observed in the volume fraction profile of the ZnO particles ( ) for T = 58 °C in Figure 6, where an initial increase is followed by reaching an almost constant value, while the vol-ume fraction of the cluster ( ) follows a sigmoidal profile. These results suggest that new ZnO particles are formed continuously throughout the reaction, feeding the clusters, which in turn grow simultaneously. Hence, at intermediate times, an equilibrium is estab-lished between ZnO particles' formation and consumption. where ∆ 2 = |ρZnO − ρMeOH | 2 . In Figure 6, the obtained cluster volume fraction ϕZnO-clus-ter,inv(t) ( ) for T = 58 °C is plotted as a function of time. Comparing the cluster volume fractions obtained from the model analysis of the USAXS data and the invariant, i.e., ϕZnO-cluster,I(q) and ϕZnO-cluster,inv, slightly lower values are found for the latter, which might be a consequence of the finite q-range used for the invariant determination as well as the ap-proximation of spherical clusters with compact packing of ZnO particles and a homoge-neous scattering length density distribution.
) with Equation (3) revealed a high q power law decay exponent of m ≈ 1.5, suggesting that the polymer chains are no longer in an ideal (m = 2) or swollen (m = 5/3) state but already approach a rod-like state (m = 1) [68,72]. A similar behavior was found by Sapir et al. [73] once the overlap concentration c* of PVP-10k in D 2 O (φ PVP4 .8 × 10 −2 ) was exceeded and the semidilute polymer regime began. This regime refers to a state in which the polymer forms a network with overlapping and entangled chains. When PVP was studied at lower volume fractions (φ PVP = 1.0, 2.0, and 4.0 × 10 −2 ) in methanol, shown in the Supplementary Materials Figures S1 and S2 and Tables S3 and S4, ideal chain behavior was observed exclusively for the lowest PVP volume fraction φ PVP = 1.0 × 10 −2 . At φ PVP = 2.0 × 10 −2 , the exponent of the power law decay drops steeply and then slowly decreases with increasing PVP amount, while R g (PVP) decreases continuously. Furthermore, the change of the scaling dependency of the correlation length, observed between φ PVP = 4.0 and 7.1 × 10 −2 , confirms that the PVP concentration in the methanolic precursor solution can be assigned to the semidilute regime [74].
The analysis of the scattering data from the methanolic PVP solution (φ PVP = 7.1 × 10 −2 ) in which ZnAc 2 ( corded slit-smeared SAXS intensities I(q) were plotted as a function q using a double-logarithmic representation. The scattering curve of PVP = 7.1 × 10 −2 (8.6 mM) ( ) shows a Guinier region at low q ttering intensity. As q increases, a power law decay is observed until t q ≈ 1 Å -1 , followed by the intermolecular C-C chain interaction peak ar behavior was found when 12.5 mM ZnAc2 ( ) or 26.3 )were added while keeping the polymer amount constant at ϕPVP r the final composition of the precursor solution ( ), ion for q < 0.1 Å −1 , a power law decay with I(q) ≈ q −1.4 can be seen. oth ZnAc2 and TEAOH results in a significantly different structure 2 and TEAOH are present) in more detail, the power law decay (I(q) low q limit of the in-house SAXS instrument provides little inforucture and size of the polymer. This information can be obtained by owards much smaller values using the USAXS/SAXS setup at the 9-vanced Photon Source at the Argonne National Laboratory [58,59]. ows a good agreement between the desmeared in-house ( ) data. In addition, the combination of the data ition of the power law into a Guinier region at q < 0.01 Å −1 . Utilizing n for low q (ln I(q) ≈ ln I0 − (Rg 2 /3) q 2 ), we obtained Rg(PVP) = 6.3 ± nds to a fourfold increase compared to Rg(PVP) in pure methanol ining only ZnAc2 or TEAOH. This significant change in the PVP increased ion concentration due to the presence of both ZnO pret likely related to a stronger repulsion of the segments within the to a further increase in the chain stiffness. This hypothesis is supin mesh size determined from the analysis at medium q's to ξPVP = , ZnAc2 might start to hydrolyze even at 25 °C, changing its chain st particle nuclei could have formed within the PVP. However, this accelerated only by increasing the temperature. umes continuous nucleation of ZnO particles. This can be indeed fraction profile of the ZnO particles ( ) for T = 58 °C in Figure 6, e is followed by reaching an almost constant value, while the volter ( ) follows a sigmoidal profile. These results suggest that new ed continuously throughout the reaction, feeding the clusters, ultaneously. Hence, at intermediate times, an equilibrium is estab-) or TEAOH ( In Figure 1a, the recorded slit-smeared SAXS intensities I(q) were plotted as a functio of the scattering vector q using a double-logarithmic representation. The scattering curve PVP in pure methanol ϕPVP = 7.1 × 10 −2 (8.6 mM) ( ) shows a Guinier region at low with almost constant scattering intensity. As q increases, a power law decay is observed un a minimum is reached at q ≈ 1 Å -1 , followed by the intermolecular C-C chain interaction pea [66] at q ≈ 1.5 Å −1 . Similar behavior was found when 12.5 mM ZnAc2 ( ) or 26 mM TEAOH ( )were added while keeping the polymer amount constant at ϕP = 7.1 × 10 −2 . However, for the final composition of the precursor solution ( instead of a Guinier region for q < 0.1 Å −1 , a power law decay with I(q) ≈ q −1.4 can be see Thus, the presence of both ZnAc2 and TEAOH results in a significantly different structu of the polymer.
(where both ZnAc2 and TEAOH are present) in more detail, the power law decay (I( ≈ q −1.4 ) recorded in the low q limit of the in-house SAXS instrument provides little info mation on the global structure and size of the polymer. This information can be obtained b extending the q-range towards much smaller values using the USAXS/SAXS setup at the ID-C beamline at the Advanced Photon Source at the Argonne National Laboratory [58,59 The inset in Figure 1 shows a good agreement between the desmeared in-house ( ) and USAXS/SAXS ( ) data. In addition, the combination of the da clearly shows the transition of the power law into a Guinier region at q < 0.01 Å −1 . Utilizin Guinier's approximation for low q (ln I(q) ≈ ln I0 − (Rg 2 /3) q 2 ), we obtained Rg(PVP) = 6.3 0.3 nm, which corresponds to a fourfold increase compared to Rg(PVP) in pure methan and in solutions containing only ZnAc2 or TEAOH. This significant change in the PV structure caused by the increased ion concentration due to the presence of both ZnO pr cursor and base is most likely related to a stronger repulsion of the segments within th polymer chain, leading to a further increase in the chain stiffness. This hypothesis is su ported by the increase in mesh size determined from the analysis at medium q's to ξPVP 2.9 ± 0.2 nm. Moreover, ZnAc2 might start to hydrolyze even at 25 °C, changing its cha flexibility so that the first particle nuclei could have formed within the PVP. However, th reaction is significantly accelerated only by increasing the temperature. The FW model assumes continuous nucleation of ZnO particles. This can be indee observed in the volume fraction profile of the ZnO particles ( ) for T = 58 °C in Figure where an initial increase is followed by reaching an almost constant value, while the vo ume fraction of the cluster ( ) follows a sigmoidal profile. These results suggest that ne ZnO particles are formed continuously throughout the reaction, feeding the cluster which in turn grow simultaneously. Hence, at intermediate times, an equilibrium is esta lished between ZnO particles' formation and consumption. ) was added revealed a slight increase in R g (PVP) (see Table 1) while the decrease in the exponent of the power law decayindicates that the polymer becomes more rod-like [72]. Though PVP is a nonionic polymer, the pyrrolidone head group is polar and interacts with the ions of the ZnO precursor and the organic base [33,75]. Their accumulation along the polymer chains, leading to repulsion of the segments within the polymer chain, can explain why the polymer chain becomes slightly stiffer and more swollen compared to the ion-free solution. This effect is slightly more pronounced with TEAOH, whose concentration is more than twice of ZnAc 2 . Table 1. Parameters obtained from the analysis of SAXS data from PVP in methanol at φ PVP = 7.1 × 10 −2 before and after adding 12.5 mM ZnAc 2 and 26.3 mM TEAOH, respectively, as well as the final composition. While the polymer with excluded volume model [67,68] provided the radius of gyration R g (PVP) and the high q power law exponent m, the exponent of the power law decay at low q m and the correlation length ξ PVP were obtained through the correlation length model [69]. Discussing the scattering data of the precursor solution ( In Figure 1a, the recorded slit-smeared SAXS intensities I(q) were plotted as a function of the scattering vector q using a double-logarithmic representation. The scattering curve of PVP in pure methanol ϕPVP = 7.1 × 10 −2 (8.6 mM) ( ) shows a Guinier region at low q with almost constant scattering intensity. As q increases, a power law decay is observed until a minimum is reached at q ≈ 1 Å -1 , followed by the intermolecular C-C chain interaction peak [66] at q ≈ 1.5 Å −1 . Similar behavior was found when 12.5 mM ZnAc2 ( ) or 26.3 mM TEAOH ( )were added while keeping the polymer amount constant at ϕPVP = 7.1 × 10 −2 . However, for the final composition of the precursor solution ( ), instead of a Guinier region for q < 0.1 Å −1 , a power law decay with I(q) ≈ q −1.4 can be seen. Thus, the presence of both ZnAc2 and TEAOH results in a significantly different structure of the polymer.
(where both ZnAc2 and TEAOH are present) in more detail, the power law decay (I(q) ≈ q −1.4 ) recorded in the low q limit of the in-house SAXS instrument provides little information on the global structure and size of the polymer. This information can be obtained by extending the q-range towards much smaller values using the USAXS/SAXS setup at the 9-ID-C beamline at the Advanced Photon Source at the Argonne National Laboratory [58,59].
The inset in Figure 1 shows a good agreement between the desmeared in-house ( ) and USAXS/SAXS ( ) data. In addition, the combination of the data clearly shows the transition of the power law into a Guinier region at q < 0.01 Å −1 . Utilizing Guinier's approximation for low q (ln I(q) ≈ ln I0 − (Rg 2 /3) q 2 ), we obtained Rg(PVP) = 6.3 ± 0.3 nm, which corresponds to a fourfold increase compared to Rg(PVP) in pure methanol and in solutions containing only ZnAc2 or TEAOH. This significant change in the PVP structure caused by the increased ion concentration due to the presence of both ZnO pre-) (where both ZnAc 2 and TEAOH are present) in more detail, the power law decay (I(q) ≈ q −1.4 ) recorded in the low q limit of the in-house SAXS instrument provides little information on the global structure and size of the polymer. This information can be obtained by extending the q-range towards much smaller values using the USAXS/SAXS setup at the 9-ID-C beamline at the Advanced Photon Source at the Argonne National Laboratory [58,59].
The inset in Figure 1 shows a good agreement between the desmeared in-house ( In Figure 1a, the recorded slit-smeared SAXS intensities I(q) were plotted as a function of the scattering vector q using a double-logarithmic representation. The scattering curve of PVP in pure methanol ϕPVP = 7.1 × 10 −2 (8.6 mM) ( ) shows a Guinier region at low q with almost constant scattering intensity. As q increases, a power law decay is observed until a minimum is reached at q ≈ 1 Å -1 , followed by the intermolecular C-C chain interaction peak [66] at q ≈ 1.5 Å −1 . Similar behavior was found when 12.5 mM ZnAc2 ( ) or 26.3 mM TEAOH ( )were added while keeping the polymer amount constant at ϕPVP = 7.1 × 10 −2 . However, for the final composition of the precursor solution ( ), instead of a Guinier region for q < 0.1 Å −1 , a power law decay with I(q) ≈ q −1.4 can be seen. Thus, the presence of both ZnAc2 and TEAOH results in a significantly different structure of the polymer.
(where both ZnAc2 and TEAOH are present) in more detail, the power law decay (I(q) ≈ q −1.4 ) recorded in the low q limit of the in-house SAXS instrument provides little information on the global structure and size of the polymer. This information can be obtained by extending the q-range towards much smaller values using the USAXS/SAXS setup at the 9-ID-C beamline at the Advanced Photon Source at the Argonne National Laboratory [58,59].
The inset in Figure 1 shows a good agreement between the desmeared in-house ( ) and USAXS/SAXS ( ) data. In addition, the combination of the data clearly shows the transition of the power law into a Guinier region at q < 0.01 Å −1 . Utilizing Guinier's approximation for low q (ln I(q) ≈ ln I0 − (Rg 2 /3) q 2 ), we obtained Rg(PVP) = 6.3 ± 0.3 nm, which corresponds to a fourfold increase compared to Rg(PVP) in pure methanol and in solutions containing only ZnAc2 or TEAOH. This significant change in the PVP structure caused by the increased ion concentration due to the presence of both ZnO precursor and base is most likely related to a stronger repulsion of the segments within the polymer chain, leading to a further increase in the chain stiffness. This hypothesis is supported by the increase in mesh size determined from the analysis at medium q's to ξPVP = ) and USAXS/SAXS ( In Figure 1a, the recorded slit-smeared SAXS intensities I(q) were plotted as a function of the scattering vector q using a double-logarithmic representation. The scattering curve of PVP in pure methanol ϕPVP = 7.1 × 10 −2 (8.6 mM) ( ) shows a Guinier region at low q with almost constant scattering intensity. As q increases, a power law decay is observed until a minimum is reached at q ≈ 1 Å -1 , followed by the intermolecular C-C chain interaction peak [66] at q ≈ 1.5 Å −1 . Similar behavior was found when 12.5 mM ZnAc2 ( ) or 26.3 mM TEAOH ( )were added while keeping the polymer amount constant at ϕPVP = 7.1 × 10 −2 . However, for the final composition of the precursor solution ( ), instead of a Guinier region for q < 0.1 Å −1 , a power law decay with I(q) ≈ q −1.4 can be seen. Thus, the presence of both ZnAc2 and TEAOH results in a significantly different structure of the polymer.
(where both ZnAc2 and TEAOH are present) in more detail, the power law decay (I(q) ≈ q −1.4 ) recorded in the low q limit of the in-house SAXS instrument provides little information on the global structure and size of the polymer. This information can be obtained by extending the q-range towards much smaller values using the USAXS/SAXS setup at the 9-ID-C beamline at the Advanced Photon Source at the Argonne National Laboratory [58,59].
The inset in Figure 1 shows a good agreement between the desmeared in-house ( ) and USAXS/SAXS ( ) data. In addition, the combination of the data clearly shows the transition of the power law into a Guinier region at q < 0.01 Å −1 . Utilizing Guinier's approximation for low q (ln I(q) ≈ ln I0 − (Rg 2 /3) q 2 ), we obtained Rg(PVP) = 6.3 ± 0.3 nm, which corresponds to a fourfold increase compared to Rg(PVP) in pure methanol and in solutions containing only ZnAc2 or TEAOH. This significant change in the PVP structure caused by the increased ion concentration due to the presence of both ZnO precursor and base is most likely related to a stronger repulsion of the segments within the polymer chain, leading to a further increase in the chain stiffness. This hypothesis is supported by the increase in mesh size determined from the analysis at medium q's to ξPVP = ) data. In addition, the combination of the data clearly shows the transition of the power law into a Guinier region at q < 0.01 Å −1 . Utilizing Guinier's approximation for low q (ln I(q) ≈ ln I 0 − (R g 2 /3) q 2 ), we obtained R g (PVP) = 6.3 ± 0.3 nm, which corresponds to a fourfold increase compared to R g (PVP) in pure methanol and in solutions containing only ZnAc 2 or TEAOH. This significant change in the PVP structure caused by the increased ion concentration due to the presence of both ZnO precursor and base is most likely related to a stronger repulsion of the segments within the polymer chain, leading to a further increase in the chain stiffness. This hypothesis is supported by the increase in mesh size determined from the analysis at medium q's to ξ PVP = 2.9 ± 0.2 nm. Moreover, ZnAc 2 might start to hydrolyze even at 25 • C, changing its chain flexibility so that the first particle nuclei could have formed within the PVP. However, this reaction is significantly accelerated only by increasing the temperature.
For the quantitative analysis of the precursor solution data, we used the flexible cylinder model for single semiflexible polymer chains [70,71], which has already been applied to the scattering of aqueous PVP-40 k and -360 k solutions [76], combined with a Gaussian to describe the intermolecular C-C chain interaction peak [66] at high q. Accordingly, the scattering intensity was modeled by where φ PVP is the volume fraction of PVP, ∆ρ 2 is the square of the scattering length density difference between PVP (ρ PVP = 11.0 × 10 −6 Å −2 ) and methanol (ρ MeOH = 7.5 × 10 −6 Å −2 ), V C is the volume of the cylinder and S(q, L, b) is the scattering of a polymer chain with excluded volume effects with the contour length L, the Kuhn length b, a measure of the local stiffness, and the circular cross section P(q, R C ) of a cylinder with the radius R C . The comprehensive mathematical expression of S(q, L, b) can be found in the work of Chen et al. [71], while P(q, R C ) is given by the first-order Bessel function J 1 with As can be seen in Figure 1, this model is able to describe the smeared and desmeared (inset) scattering data of the precursor solution, almost quantitatively, yielding a contour length of L = 80 ± 1 nm, a Kuhn length of b = 3.3 ± 0.3 nm, and a cross-section radius of R C = 0.27 ± 0.01 nm. b and R C are in the same range, as already reported for PVP-40 k and -360 k in dilute aqueous solutions (b = 3.0 ± 0.2 nm and R C ≈ 0.1 nm [76]) as well as for PVP-2 k to -2200 k in 0.1 M aqueous NaAc solution (b ≈ 1.7 − 2.4 nm and R C = 0.25 ± 0.01 nm [77]). Estimating the radius of gyration from the contour and Kuhn length with R g 2 ≈ Lb/6 [78], we obtain R g (PVP) = 6.5 ± 0.4 nm, in good agreement with the Guinier analysis. Note that in the region around q~0.1 Å, the modeled scattering curve shows a stronger curvature than the recorded data. This small but systematic difference is most likely related to polymer-polymer interactions, which are neglected in the applied model [70]. The shape-and size-independent invariant analysis of the desmeared USAXS/SAXS data (q < 1 Å −1 ), which are on absolute scale, has shown that the scattering length density difference ∆ρ had to be increased from ∆ρ = ρ PVP − ρ MeOH = 3.5 × 10 −6 Å −2 to ∆ρ = 5.6 × 10 −6 Å −2 in order to obtain the experimental PVP volume fraction. The same increase in contrast factor was also obtained from the analysis of desmeared in-house SAXS data as well as of additional SAXS data recorded at the ID02 beamline at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France [79]. The higher X-ray scattering contrast might be related to ion association along the polymer chain as well as additional specific volume effects, which both increase the scattering length density of the polymer.
In summary, due to the relatively high volume fraction of PVP, the initial state of the precursor solution is characterized by the scattering of a PVP network. Ions are distributed along and within the cavities of the PVP network, where they interact with the pyrrolidone moiety of the polymer, leading to a semiflexible polymer chain behavior.

In Situ Study of ZnO Particle Formation and Growth
Having studied the initial state of the precursor solution at 25 • C, we then focused on elucidating the formation of ZnO particles induced by heating. Typically, the reaction is carried out at 60 • C, as ZnAc 2 hydrolyzation and mass transport during the ZnO formation are accelerated at this temperature. Herein we chose to study the ZnO kinetics at T = 58 • C and 63 • C. The reason for that is the significantly slower deposition below 50 • C, as confirmed in preliminary studies, while the upper limitation for the reaction temperature is given by the boiling point of methanol at 65 • C. Figure 2 shows the timedependent slit-smeared USAXS curves starting from 25 • C (t = 0 min) followed by rapid heating to T = 58 • C (a) or 63 • C (b).

In Situ Study of ZnO Particle Formation and Growth
Having studied the initial state of the precursor solution at 25 °C, we then focused on elucidating the formation of ZnO particles induced by heating. Typically, the reaction is carried out at 60 °C, as ZnAc2 hydrolyzation and mass transport during the ZnO formation are accelerated at this temperature. Herein we chose to study the ZnO kinetics at T = 58 °C and 63 °C. The reason for that is the significantly slower deposition below 50 °C, as confirmed in preliminary studies, while the upper limitation for the reaction temperature is given by the boiling point of methanol at 65 °C. Figure 2 shows the time-dependent slitsmeared USAXS curves starting from 25 °C (t = 0 min) followed by rapid heating to T = 58 °C (a) or 63 °C (b). During heating and temperature equilibration the precursor solution to T = 58 °C (Figure 2a), which took about 10 min, the scattering curves are dominated by the scattering of the polymer network with the above-mentioned features of a Guinier region at low q, followed by a power law decay. The slight decrease in the intensity at low q might relate to the heating with an inhomogeneous temperature distribution inside the solution. Potential over subtraction of solvent scattering, which was measured at T = 58 °C, might be another explanation for this decrease.
Subsequently, for reaction times t > 13 min, the scattering intensities at low q increase continuously, which can be attributed to the form factor contribution of newly formed ZnO species, while above q ≥ 0.1 Å −1 , the scattering is dominated by the PVP scattering. As the reaction progresses, the form factor minimum becomes increasingly apparent and shifts to lower q values, suggesting the growth of the ZnO species. Finally, the increase in the low q scattering intensity levels off, indicating that the precursor ZnAc2 has been consumed; therefore, the data acquisition was stopped after 87 min. Contrary to what Lipowsky et al. [33] have seen when ZnO films are formed in the presence of PVP via CBD, no additional contribution of needle-like species with a length scale between 20 and 50 nm was observed in the scattering curves.
A similar behavior of the scattering curves can be seen during the formation of ZnO particles at T = 63 °C in Figure 2b. Compared to the formation at 58 °C, the decrease in the During heating and temperature equilibration the precursor solution to T = 58 • C (Figure 2a), which took about 10 min, the scattering curves are dominated by the scattering of the polymer network with the above-mentioned features of a Guinier region at low q, followed by a power law decay. The slight decrease in the intensity at low q might relate to the heating with an inhomogeneous temperature distribution inside the solution. Potential over subtraction of solvent scattering, which was measured at T = 58 • C, might be another explanation for this decrease.
Subsequently, for reaction times t > 13 min, the scattering intensities at low q increase continuously, which can be attributed to the form factor contribution of newly formed ZnO species, while above q ≥ 0.1 Å −1 , the scattering is dominated by the PVP scattering. As the reaction progresses, the form factor minimum becomes increasingly apparent and shifts to lower q values, suggesting the growth of the ZnO species. Finally, the increase in the low q scattering intensity levels off, indicating that the precursor ZnAc 2 has been consumed; therefore, the data acquisition was stopped after 87 min. Contrary to what Lipowsky et al. [33] have seen when ZnO films are formed in the presence of PVP via CBD, no additional contribution of needle-like species with a length scale between 20 and 50 nm was observed in the scattering curves.
A similar behavior of the scattering curves can be seen during the formation of ZnO particles at T = 63 • C in Figure 2b. Compared to the formation at 58 • C, the decrease in the forward scattering observed in the curves recorded at the early stages of the process is even more pronounced, most likely due to over subtraction of the background, measured at the higher final temperature. The earlier appearance of the form factor signature and the earlier leveling off of the increase in forward scattering after 75 min clearly indicate that the reaction is faster at higher temperatures. This can be explained by the enhanced mass transport with increasing temperature, which accelerates the formation of ZnO.
A comparison of the first (t = 0 min, T = 25 • C) and the last (t = 87 min, T = 58 • C) timeresolved USAXS/SAXS scattering curves are shown by means of the ZnO particles' formation at T = 58 • C (Figure 3, left). This comparison highlights the presence of the additional form factor contribution (q ≤ 0.08 Å −1 ) in the scattering curve recorded at the process end (t = 87 min). A closer look also reveals a weak shoulder around q~0.15 Å −1 . One exemplary image of complementary TEM studies taken from a separately prepared solution after an analogous heating procedure starting from 25 • C to 58 • C is shown in the center of Figure 3. Therein, two species are visible: smaller ones with a radius of R ZnO,TEM = 1.3 ± 0.7 nm as well as agglomerates of those with an average radius of R ZnO-cluster,TEM = 8.5 ± 1.4 nm. We will refer to the smaller particles as individual ZnO particles and to the larger ones as clusters of ZnO particles. Strikingly, the diameter of the individual ZnO particles is only slightly smaller than the mesh size within the PVP network (ξ PVP ≈ 2.9 nm), indicating initial growth of the ZnO particles in the network cavities before cluster formation occurs. Furthermore, the average cluster radius is only slightly larger than the Guinier radius of PVP in the precursor solution with R g (PVP) = 6.3 ± 0.3 nm. even more pronounced, most likely due to over subtraction of the background, measured at the higher final temperature. The earlier appearance of the form factor signature and the earlier leveling off of the increase in forward scattering after 75 min clearly indicate that the reaction is faster at higher temperatures. This can be explained by the enhanced mass transport with increasing temperature, which accelerates the formation of ZnO.
A comparison of the first (t = 0 min, T = 25 °C) and the last (t = 87 min, T = 58 °C) timeresolved USAXS/SAXS scattering curves are shown by means of the ZnO particles' formation at T = 58 °C (Figure 3, left). This comparison highlights the presence of the additional form factor contribution (q ≤ 0.08 Å −1 ) in the scattering curve recorded at the process end (t = 87 min). A closer look also reveals a weak shoulder around q~0.15 Å −1 . One exemplary image of complementary TEM studies taken from a separately prepared solution after an analogous heating procedure starting from 25 °C to 58 °C is shown in the center of Figure 3. Therein, two species are visible: smaller ones with a radius of RZnO,TEM = 1.3 ± 0.7 nm as well as agglomerates of those with an average radius of RZnO-cluster,TEM = 8.5 ± 1.4 nm. We will refer to the smaller particles as individual ZnO particles and to the larger ones as clusters of ZnO particles. Strikingly, the diameter of the individual ZnO particles is only slightly smaller than the mesh size within the PVP network (ξPVP ≈ 2.9 nm), indicating initial growth of the ZnO particles in the network cavities before cluster formation occurs. Furthermore, the average cluster radius is only slightly larger than the Guinier radius of PVP in the precursor solution with Rg(PVP) = 6.3 ± 0.3 nm.  [70,71] used for t = 0 min was additively combined with two polydisperse spherical form factors [80] for the scattering contributions of the ZnO particles and their clusters, as well as with a Gaussian. Right: TEM image taken after the ZnO synthesis at T = 58 °C with the corresponding volume-weighted size distribution of the ZnO particles and clusters, described by a Gaussian size distribution, respectively.
Comparing the final cluster radius obtained from the USAXS/SAXS scattering curve analysis (RZnO-cluster,f = 10.5 ± 1.9 nm) to TEM, one may assume that the clustered particles are surrounded by a PVP layer that is not visible in the TEM, similar to what has been demonstrated for silica-polymethyl methacrylate (PMMA) nanocomposites by SANS [81]. Since we were not able to study such a layer independently with SANS, we used a simple approach for the quantitative analysis of the time-resolved USAXS/SAXS data. In this approach, the scattering contribution of the PVP network described by the flexible cylinder model [70,71] I(q)flexC was additively combined with the scattering contributions of the  [70,71] used for t = 0 min was additively combined with two polydisperse spherical form factors [80] for the scattering contributions of the ZnO particles and their clusters, as well as with a Gaussian. Right: TEM image taken after the ZnO synthesis at T = 58 • C with the corresponding volume-weighted size distribution of the ZnO particles and clusters, described by a Gaussian size distribution, respectively.
Comparing the final cluster radius obtained from the USAXS/SAXS scattering curve analysis (R ZnO-cluster,f = 10.5 ± 1.9 nm) to TEM, one may assume that the clustered particles are surrounded by a PVP layer that is not visible in the TEM, similar to what has been demonstrated for silica-polymethyl methacrylate (PMMA) nanocomposites by SANS [81]. Since we were not able to study such a layer independently with SANS, we used a simple approach for the quantitative analysis of the time-resolved USAXS/SAXS data. In this approach, the scattering contribution of the PVP network described by the flexible cylinder model [70,71] I(q) flexC was additively combined with the scattering contributions of the individual ZnO particles I(q) ZnO and their clusters I(q) ZnO-cluster , each modeled by the form factor of polydisperse spheres [80], as well as a Gaussian I(q) G for the intermolecular C-C chain interaction peak [66] at high q. Thus, the total intensity is given by I(q) = I(q) flexC + I(q) ZnO + I(q) ZnO−cluster + I(q) G + I background (6) where with φ ZnO,i and V ZnO,i being the volume fractions and average volumes of particles and particle clusters, respectively. ∆ρ 2 is the scattering length density difference between ZnO (ρ ZnO = 44.8 × 10 −6 Å −2 ) and methanol (ρ MeOH = 7.5 × 10 −6 Å −2 ) and P(q,R ZnO,i ) = 3J 1 (qR ZnO,i ) qR ZnO,i 2 the form factor of a sphere with the radius R ZnO,i . To account for size distribution effects, the form factor was convoluted with a Gaussian size distribution functions f (R ZnO,i , R ZnO,i,f , σ i ) which yields the mean radius R ZnO,i,f and the width σ i .
Using this approach, we were able to describe the recorded time-resolved scattering curves almost quantitatively, as shown in the left panel of Figure 3. In this regard, the solid line for t = 0 min corresponds to the final fit following Equation (5), with a PVP chain contour length of L = 80 ± 1 nm, a Kuhn length of b = 3.3 ± 0.3 nm, and a cross-section radius of R C = 0.27 ± 0.01 nm. For the solid line at t = 87 min (T = 58 • C), Equation (6) was used. Here, the previously determined characteristics of the PVP chains with a slightly adjusted Kuhn length of b = 4.0 ± 0.3 nm were applied, while the polydispersity p i = σ i /R ZnO,i0 = 0.18 of both the ZnO particles and clusters was kept constant. Hence, the analysis of the USAXS/SAXS curve measured after the termination of the reaction yields average radii of R ZnO,f = 1.5 ± 0.3 nm and R ZnO-cluster,f = 10.5 ± 1.9 nm for the individual ZnO particles and their clusters, respectively. While the size of the individual ZnO particles matches that from the TEM images within the measurement error, the size of the clusters is slightly larger, most probably due to a surrounding PVP layer, as discussed before. The respective contributions of Equation (6) to the final fit are shown in Figure S3 in the Supplementary Materials.
Also, for the ZnO formation performed at T = 63 • C, starting from T = 25 • C, the final ZnO particle and cluster size obtained from TEM images (R ZnO,TEM = 1.7 ± 0.5 nm, R ZnO-cluster,TEM = 9.7 ± 1.9 nm), shown exemplarily in Figure S4 in the Supplementary Materials, are in good accordance with that from USAXS/SAXS (R ZnO,f = 1.5 ± 0.3 nm, R ZnO-cluster,f = 11.8 ± 2.1 nm). Comparing the size of the clusters obtained from CBD at the two different temperatures, the temperature increase leads to slightly larger ZnO clusters, which is related to accelerated kinetics at higher temperatures. Hereby, a higher number of individual ZnO particles might be formed, which are subsequently consumed faster by the formation of the clusters. Figure 4 shows selected time-resolved USAXS scattering curves recorded at T = 58 • C ( Figure 4a) and T = 63 • C (Figure 4b), which are almost quantitatively described by the combination of the flexible cylinder model [70,71] and the two spherical form factor contributions [80]. It should be noted that for t < 13 min, we could not obtain any information about the ZnO formation due to the dominance of the PVP scattering contribution.  The results of the quantitative analysis of the scattering data according to Equation (6), with the free fitting parameters being the mean radii from the ZnO particles, RZnO,f, and that from the clusters RZnO,cluster,f as well as their volume fractions ϕZnO and ϕZnO-cluster, are  The results of the quantitative analysis of the scattering data according to Equation (6), with the free fitting parameters being the mean radii from the ZnO particles, R ZnO,f , and that from the clusters R ZnO,cluster,f as well as their volume fractions φ ZnO and φ ZnO-cluster , are shown in Figures 5 and 6. While Figure 5 shows the radius (R) and R 3 (t) of the ZnO nanoparticles and ZnO clusters as a function of time for T = 58 • C and T = 63 • C, the time evolution of the respective volume fraction for T = 58 • C is shown in Figure 6. Interestingly, when plotting R 3 ZnO-cluster,f as a function of time, as shown in Figure 5b, a linear increase starting from t~20 min is observed, which levels off to a plateau. The former region can be associated with the growth phase of the ZnO clusters by agglomeration of individual ZnO particles. Thus, a reasonable description of the growth phase is given by with k being the temperature-dependent growth rate constant, and tcluster being the time after which clusters start to grow tcluster = t − tcluster,0 and RZnO,f the individual ZnO nanoparticle radius. The analysis of R 3 ZnO-cluster,f(t) with Equation (8)  For comparison, much smaller rate constants in the order of ~10 −3 nm 3 /s were found for the formation of ZnO particles in propanol, without the addition of a polymer such as PVP [82,83]. The large growth rates obtained here can be explained by the occurrence of depletion forces [84,85], as observed during ZnS particle formation in the presence of PVP, where cluster growth via depletion forces appeared as soon as the particle surface was too crowded by PVP [86].
While large parts of the ZnO cluster growth phase can reasonably be described by this power law approach, neither the nucleation phase nor the growth deceleration can be described. Furthermore, the occurrence of depletion forces is not considered in this model. Thus, to describe the sigmoidal time evolution of R 3 (t), the two-step Finke-Watzky (FW) model was used. In this model slow, continuous nucleation is followed by autocatalytic growth + 2 with the nucleation (k1) and growth (k2) constants [62][63][64]. Here, the cube of the radius as a function of time is given by [65] ≥ ( ) = · 1 − + · ( · ) (9) In Figure 1a, the recorded slit-smeared SAXS intensities I(q) were plotted as a function of the scattering vector q using a double-logarithmic representation. The scattering curve of PVP in pure methanol ϕPVP = 7.1 × 10 −2 (8.6 mM) ( ) shows a Guinier region at low q with almost constant scattering intensity. As q increases, a power law decay is observed until a minimum is reached at q ≈ 1 Å -1 , followed by the intermolecular C-C chain interaction peak [66] at q ≈ 1.5 Å −1 . Similar behavior was found when 12.5 mM ZnAc2 ( ) or 26.3 mM TEAOH ( )were added while keeping the polymer amount constant at ϕPVP = 7.1 × 10 −2 . However, for the final composition of the precursor solution ( ), instead of a Guinier region for q < 0.1 Å −1 , a power law decay with I(q) ≈ q −1.4 can be seen. Thus, the presence of both ZnAc2 and TEAOH results in a significantly different structure of the polymer.
(where both ZnAc2 and TEAOH are present) in more detail, the power law decay (I(q) ≈ q −1.4 ) recorded in the low q limit of the in-house SAXS instrument provides little information on the global structure and size of the polymer. This information can be obtained by extending the q-range towards much smaller values using the USAXS/SAXS setup at the 9-ID-C beamline at the Advanced Photon Source at the Argonne National Laboratory [58,59].
The inset in Figure 1 shows a good agreement between the desmeared in-house ( ) and USAXS/SAXS ( ) data. In addition, the combination of the data clearly shows the transition of the power law into a Guinier region at q < 0.01 Å −1 . Utilizing Guinier's approximation for low q (ln I(q) ≈ ln I0 − (Rg 2 /3) q 2 ), we obtained Rg(PVP) = 6.3 ± 0.3 nm, which corresponds to a fourfold increase compared to Rg(PVP) in pure methanol and in solutions containing only ZnAc2 or TEAOH. This significant change in the PVP structure caused by the increased ion concentration due to the presence of both ZnO precursor and base is most likely related to a stronger repulsion of the segments within the polymer chain, leading to a further increase in the chain stiffness. This hypothesis is supported by the increase in mesh size determined from the analysis at medium q's to ξPVP = 2.9 ± 0.2 nm. Moreover, ZnAc2 might start to hydrolyze even at 25 °C, changing its chain flexibility so that the first particle nuclei could have formed within the PVP. However, this reaction is significantly accelerated only by increasing the temperature. The FW model assumes continuous nucleation of ZnO particles. This can be indeed observed in the volume fraction profile of the ZnO particles ( ) for T = 58 °C in Figure 6, where an initial increase is followed by reaching an almost constant value, while the volume fraction of the cluster ( ) follows a sigmoidal profile. These results suggest that new ZnO particles are formed continuously throughout the reaction, feeding the clusters, which in turn grow simultaneously. Hence, at intermediate times, an equilibrium is established between ZnO particles' formation and consumption. where ∆ 2 = |ρZnO − ρMeOH | 2 . In Figure 6, the obtained cluster volume fraction ϕZnO-cluster,inv(t) ( ) for T = 58 °C is plotted as a function of time. Comparing the cluster volume fractions obtained from the model analysis of the USAXS data and the invariant, i.e., ϕZnOcluster,I(q) and ϕZnO-cluster,inv, slightly lower values are found for the latter, which might be a consequence of the finite q-range used for the invariant determination as well as the approximation of spherical clusters with compact packing of ZnO particles and a homogeneous scattering length density distribution. In Figure 1a, the recorded slit-smeared SAXS intensities I(q) were plotted as a function of the scattering vector q using a double-logarithmic representation. The scattering curve of PVP in pure methanol ϕPVP = 7.1 × 10 −2 (8.6 mM) ( ) shows a Guinier region at low q with almost constant scattering intensity. As q increases, a power law decay is observed until a minimum is reached at q ≈ 1 Å -1 , followed by the intermolecular C-C chain interaction peak [66] at q ≈ 1.5 Å −1 . Similar behavior was found when 12.5 mM ZnAc2 ( ) or 26.3 mM TEAOH ( )were added while keeping the polymer amount constant at ϕPVP = 7.1 × 10 −2 . However, for the final composition of the precursor solution ( ), instead of a Guinier region for q < 0.1 Å −1 , a power law decay with I(q) ≈ q −1.4 can be seen. Thus, the presence of both ZnAc2 and TEAOH results in a significantly different structure of the polymer.
(where both ZnAc2 and TEAOH are present) in more detail, the power law decay (I(q) ≈ q −1.4 ) recorded in the low q limit of the in-house SAXS instrument provides little information on the global structure and size of the polymer. This information can be obtained by extending the q-range towards much smaller values using the USAXS/SAXS setup at the 9-ID-C beamline at the Advanced Photon Source at the Argonne National Laboratory [58,59].
The inset in Figure 1 shows a good agreement between the desmeared in-house ( ) and USAXS/SAXS ( ) data. In addition, the combination of the data clearly shows the transition of the power law into a Guinier region at q < 0.01 Å −1 . Utilizing Guinier's approximation for low q (ln I(q) ≈ ln I0 − (Rg 2 /3) q 2 ), we obtained Rg(PVP) = 6.3 ± 0.3 nm, which corresponds to a fourfold increase compared to Rg(PVP) in pure methanol and in solutions containing only ZnAc2 or TEAOH. This significant change in the PVP structure caused by the increased ion concentration due to the presence of both ZnO precursor and base is most likely related to a stronger repulsion of the segments within the polymer chain, leading to a further increase in the chain stiffness. This hypothesis is supported by the increase in mesh size determined from the analysis at medium q's to ξPVP = 2.9 ± 0.2 nm. Moreover, ZnAc2 might start to hydrolyze even at 25 °C, changing its chain flexibility so that the first particle nuclei could have formed within the PVP. However, this reaction is significantly accelerated only by increasing the temperature. The FW model assumes continuous nucleation of ZnO particles. This can be indeed observed in the volume fraction profile of the ZnO particles ( ) for T = 58 °C in Figure 6, where an initial increase is followed by reaching an almost constant value, while the volume fraction of the cluster ( ) follows a sigmoidal profile. These results suggest that new ZnO particles are formed continuously throughout the reaction, feeding the clusters, which in turn grow simultaneously. Hence, at intermediate times, an equilibrium is established between ZnO particles' formation and consumption. where ∆ 2 = |ρZnO − ρMeOH | 2 . In Figure 6, the obtained cluster volume fraction ϕZnO-cluster,inv(t) ( ) for T = 58 °C is plotted as a function of time. Comparing the cluster volume fractions obtained from the model analysis of the USAXS data and the invariant, i.e., ϕZnOcluster,I(q) and ϕZnO-cluster,inv, slightly lower values are found for the latter, which might be a consequence of the finite q-range used for the invariant determination as well as the approximation of spherical clusters with compact packing of ZnO particles and a homogeneous scattering length density distribution.  In a complementary approach, we determined the volume fraction of ZnO clusters ϕZnO-cluster from the shape-and size-independent scattering invariant Q of the time-resolved desmeared USAXS scattering curves for q ≤ 0.04 Å −1 . It should be noted that the invariant was only determined in the low q-part, since on the one hand, the scattering in the large q-part is dominated by the scattering of PVP, and on the other hand, the USAXS data are noisy in this range. To account only for the scattering of the ZnO clusters, the scattering invariant determined for t = 1 min was subtracted from all other data, resulting in a reduced invariant Q*, according to As can be seen in Figure 5a, the size of the formed ZnO species for T = 58 • C remains almost constant during a short period (~between t = 13 and 18 min). Thereafter, cluster growth occurs, where the average size of these clusters initially increases rapidly before slowly approaching the final size due to the consumption of zinc precursor. Performing the reaction at T = 63 • C, a similar but steeper growth profile was obtained, during which the clusters are consistently larger. For comparison, the ZnO particle size appeared to remain constant and was kept fixed at R ZnO,f = 1.5 ± 0.3 nm (dashed line in Figure 5a).
Interestingly, when plotting R 3 ZnO-cluster,f as a function of time, as shown in Figure 5b, a linear increase starting from t~20 min is observed, which levels off to a plateau. The former region can be associated with the growth phase of the ZnO clusters by agglomeration of individual ZnO particles. Thus, a reasonable description of the growth phase is given by with k being the temperature-dependent growth rate constant, and t cluster being the time after which clusters start to grow t cluster = t − t cluster,0 and R ZnO,f the individual ZnO nanoparticle radius. The analysis of R 3 ZnO-cluster,f (t) with Equation (8) (dotted lines in Figure 5b) yielded k(T = 58 • C) = 0.44 ± 0.01 nm 3 /s with t cluster,0 (T = 58 • C) = 23 min and k(T = 63 • C) = 1.05 ± 0.01 nm 3 /s with t cluster,0 (T = 63 • C) = 18 min. For comparison, much smaller rate constants in the order of~10 −3 nm 3 /s were found for the formation of ZnO particles in propanol, without the addition of a polymer such as PVP [82,83].
The large growth rates obtained here can be explained by the occurrence of depletion forces [84,85], as observed during ZnS particle formation in the presence of PVP, where cluster growth via depletion forces appeared as soon as the particle surface was too crowded by PVP [86].
While large parts of the ZnO cluster growth phase can reasonably be described by this power law approach, neither the nucleation phase nor the growth deceleration can be described. Furthermore, the occurrence of depletion forces is not considered in this model. Thus, to describe the sigmoidal time evolution of R 3 (t), the two-step Finke-Watzky (FW) model was used. In this model slow, continuous nucleation A k 1 → B is followed by autocatalytic growth A + B k 2 → 2B with the nucleation (k 1 ) and growth (k 2 ) constants [62][63][64]. Here, the cube of the radius as a function of time is given by [65] where R f corresponds to the final cluster size, and [A] 0 to the starting concentration of Zn 2+ . The induction time, here related to the time after which agglomeration of the ZnO particles into clusters is accelerated, is defined by the intersection of the initial and maximal slope and can be determined according to [65]: Furthermore, the ratio of the growth to nucleation rate constant S = k 2 k 1 (M −1 ) is a measure of kinetic control, indicating whether nucleation of new particles or growth of the existing clusters is preferred. Thus, larger S values yield narrow size distributions. In addition, larger ratios were shown to be related to larger clusters [87]. Note that the two-step FW model can be extended to include bimolecular aggregation (B + B→C) [88] and autocatalytic agglomeration (B + C→1.5 C) [89,90]. However, this was not performed here to keep the number of free-fitting parameters at a minimum.
As can be seen from the solid lines in Figure 5, the FW model provides a quantitative description of the time evolution of the ZnO cluster size for t ≥ t ind . The systematic mismatch of the model at t < t ind could be due, on one hand, to a nucleation rate that is not constant during the heating process but increases strongly. On the other hand, the spherical form factor used in the analysis of the scattering data represents only an approximation, especially for clusters consisting of a few ZnO particles. The obtained parameters, summarized in Table 2, reflect what was already evident from the USAXS data: the induction time shortens from t ind = 27 min at T = 58 • C to t ind = 20 min for T = 63 • C, while the growth rate constant k 2 increases considerably by a factor of 50% at higher temperatures to k 2 [A] 0 = 0.17 s −1 . The nucleation rate, however, does not change. Consequently, the ratio of the growth-to-nucleation rate constant S is higher for T = 63 • C, which also confirms the previously mentioned assumption of a faster growth rate for higher temperatures. Table 2. Values for the nucleation rate k 1 and growth rate constant k 2 [A] 0 determined from the two-step Finke-Watzky model [62][63][64][65] together with the induction time t ind and the ratio of the rate constants S obtained from the size evolution profile R 3 (t) FW and the volume fractions from the invariant analysis φ ZnO-cluster, inv (t) FW . We estimate the relative errors to be about 10%. Thus, the presence of both ZnAc2 and TEAOH results in a significantly different structure of the polymer. (where both ZnAc2 and TEAOH are present) in more detail, the power law decay (I(q) ≈ q −1.4 ) recorded in the low q limit of the in-house SAXS instrument provides little information on the global structure and size of the polymer. This information can be obtained by extending the q-range towards much smaller values using the USAXS/SAXS setup at the 9-ID-C beamline at the Advanced Photon Source at the Argonne National Laboratory [58,59].
The inset in Figure 1 shows a good agreement between the desmeared in-house ( ) and USAXS/SAXS ( ) data. In addition, the combination of the data clearly shows the transition of the power law into a Guinier region at q < 0.01 Å −1 . Utilizing Guinier's approximation for low q (ln I(q) ≈ ln I0 − (Rg 2 /3) q 2 ), we obtained Rg(PVP) = 6.3 ± 0.3 nm, which corresponds to a fourfold increase compared to Rg(PVP) in pure methanol and in solutions containing only ZnAc2 or TEAOH. This significant change in the PVP structure caused by the increased ion concentration due to the presence of both ZnO precursor and base is most likely related to a stronger repulsion of the segments within the polymer chain, leading to a further increase in the chain stiffness. This hypothesis is supported by the increase in mesh size determined from the analysis at medium q's to ξPVP = 2.9 ± 0.2 nm. Moreover, ZnAc2 might start to hydrolyze even at 25 °C, changing its chain flexibility so that the first particle nuclei could have formed within the PVP. However, this reaction is significantly accelerated only by increasing the temperature. The FW model assumes continuous nucleation of ZnO particles. This can be indeed observed in the volume fraction profile of the ZnO particles ( ) for T = 58 °C in Figure 6, where an initial increase is followed by reaching an almost constant value, while the volume fraction of the cluster ( ) follows a sigmoidal profile. These results suggest that new ZnO particles are formed continuously throughout the reaction, feeding the clusters, which in turn grow simultaneously. Hence, at intermediate times, an equilibrium is established between ZnO particles' formation and consumption. where ∆ 2 = |ρZnO − ρMeOH | 2 . In Figure 6, the obtained cluster volume fraction ϕZnO-cluster,inv(t) ( ) for T = 58 °C is plotted as a function of time. Comparing the cluster volume fractions obtained from the model analysis of the USAXS data and the invariant, i.e., ϕZnOcluster,I(q) and ϕZnO-cluster,inv, slightly lower values are found for the latter, which might be a consequence of the finite q-range used for the invariant determination as well as the approximation of spherical clusters with compact packing of ZnO particles and a homogeneous scattering length density distribution.
) for T = 58 • C in Figure 6, where an initial increase is followed by reaching an almost constant value, while the volume fraction of the cluster ( Thus, the presence of both ZnAc2 and TEAOH results in a significantly different structure of the polymer. (where both ZnAc2 and TEAOH are present) in more detail, the power law decay (I(q) ≈ q −1.4 ) recorded in the low q limit of the in-house SAXS instrument provides little information on the global structure and size of the polymer. This information can be obtained by extending the q-range towards much smaller values using the USAXS/SAXS setup at the 9-ID-C beamline at the Advanced Photon Source at the Argonne National Laboratory [58,59].
The inset in Figure 1 shows a good agreement between the desmeared in-house ( ) and USAXS/SAXS ( ) data. In addition, the combination of the data clearly shows the transition of the power law into a Guinier region at q < 0.01 Å −1 . Utilizing Guinier's approximation for low q (ln I(q) ≈ ln I0 − (Rg 2 /3) q 2 ), we obtained Rg(PVP) = 6.3 ± 0.3 nm, which corresponds to a fourfold increase compared to Rg(PVP) in pure methanol and in solutions containing only ZnAc2 or TEAOH. This significant change in the PVP structure caused by the increased ion concentration due to the presence of both ZnO precursor and base is most likely related to a stronger repulsion of the segments within the polymer chain, leading to a further increase in the chain stiffness. This hypothesis is supported by the increase in mesh size determined from the analysis at medium q's to ξPVP = 2.9 ± 0.2 nm. Moreover, ZnAc2 might start to hydrolyze even at 25 °C, changing its chain flexibility so that the first particle nuclei could have formed within the PVP. However, this reaction is significantly accelerated only by increasing the temperature. The FW model assumes continuous nucleation of ZnO particles. This can be indeed observed in the volume fraction profile of the ZnO particles ( ) for T = 58 °C in Figure 6, where an initial increase is followed by reaching an almost constant value, while the vol-ume fraction of the ) follows a sigmoidal profile. These results suggest that new ZnO particles are formed continuously throughout the reaction, feeding the clusters, which in turn grow simultaneously. Hence, at intermediate times, an equilibrium is estab-lished between ZnO particles' formation and consumption. where ∆ 2 = |ρZnO − ρMeOH | 2 . In Figure 6, the obtained cluster volume fraction ϕZnO-cluster,inv(t) ( ) for T = 58 °C is plotted as a function of time. Comparing the cluster volume fractions obtained from the model analysis of the USAXS data and the invariant, i.e., ϕZnOcluster,I(q) and ϕZnO-cluster,inv, slightly lower values are found for the latter, which might be a consequence of the finite q-range used for the invariant determination as well as the approximation of spherical clusters with compact packing of ZnO particles and a homogeneous scattering length density distribution.
) follows a sigmoidal profile. These results suggest that new ZnO particles are formed continuously throughout the reaction, feeding the clusters, which in turn grow simultaneously. Hence, at intermediate times, an equilibrium is established between ZnO particles' formation and consumption.
Since R 3 is proportional to the volume fraction, Equation (9) can also be used to model φ ZnO-cluster (t) I(q) ( Nanomaterials 2023, 13, x. https://doi.org/10.3390/xxxxx www.mdpi.com/journal/nanomaterials a minimum is reached at q ≈ 1 Å -1 , followed by the intermolecular C-C chain interaction peak [66] at q ≈ 1.5 Å −1 . Similar behavior was found when 12.5 mM ZnAc2 ( ) or 26.3 mM TEAOH ( )were added while keeping the polymer amount constant at ϕPVP = 7.1 × 10 −2 . However, for the final composition of the precursor solution ( ), instead of a Guinier region for q < 0.1 Å −1 , a power law decay with I(q) ≈ q −1.4 can be seen. Thus, the presence of both ZnAc2 and TEAOH results in a significantly different structure of the polymer.
(where both ZnAc2 and TEAOH are present) in more detail, the power law decay (I(q) ≈ q −1.4 ) recorded in the low q limit of the in-house SAXS instrument provides little information on the global structure and size of the polymer. This information can be obtained by extending the q-range towards much smaller values using the USAXS/SAXS setup at the 9-ID-C beamline at the Advanced Photon Source at the Argonne National Laboratory [58,59].
The inset in Figure 1 shows a good agreement between the desmeared in-house ( ) and USAXS/SAXS ( ) data. In addition, the combination of the data clearly shows the transition of the power law into a Guinier region at q < 0.01 Å −1 . Utilizing Guinier's approximation for low q (ln I(q) ≈ ln I0 − (Rg 2 /3) q 2 ), we obtained Rg(PVP) = 6.3 ± 0.3 nm, which corresponds to a fourfold increase compared to Rg(PVP) in pure methanol and in solutions containing only ZnAc2 or TEAOH. This significant change in the PVP structure caused by the increased ion concentration due to the presence of both ZnO precursor and base is most likely related to a stronger repulsion of the segments within the polymer chain, leading to a further increase in the chain stiffness. This hypothesis is supported by the increase in mesh size determined from the analysis at medium q's to ξPVP = 2.9 ± 0.2 nm. Moreover, ZnAc2 might start to hydrolyze even at 25 °C, changing its chain flexibility so that the first particle nuclei could have formed within the PVP. However, this reaction is significantly accelerated only by increasing the temperature. The FW model assumes continuous nucleation of ZnO particles. This can be indeed observed in the volume fraction profile of the ZnO particles ( ) for T = 58 °C in Figure 6, where an initial increase is followed by reaching an almost constant value, while the vol-ume fraction of the ) follows a sigmoidal profile. These results suggest that new ZnO particles are formed continuously throughout the reaction, feeding the clusters, which in turn grow simultaneously. Hence, at intermediate times, an equilibrium is estab-lished between ZnO particles' formation and consumption. where ∆ 2 = |ρZnO − ρMeOH | 2 . In Figure 6, the obtained cluster volume fraction ϕZnO-cluster,inv(t) ( ) for T = 58 °C is plotted as a function of time. Comparing the cluster volume fractions obtained from the model analysis of the USAXS data and the invariant, i.e., ϕZnOcluster,I(q) and ϕZnO-cluster,inv, slightly lower values are found for the latter, which might be a consequence of the finite q-range used for the invariant determination as well as the approximation of spherical clusters with compact packing of ZnO particles and a homogeneous scattering length density distribution.
In a complementary approach, we determined the volume fraction of ZnO clusters φ ZnO-cluster from the shape-and size-independent scattering invariant Q of the time-resolved desmeared USAXS scattering curves for q ≤ 0.04 Å −1 . It should be noted that the invariant was only determined in the low q-part, since on the one hand, the scattering in the large q-part is dominated by the scattering of PVP, and on the other hand, the USAXS data are noisy in this range. To account only for the scattering of the ZnO clusters, the scattering invariant determined for t = 1 min was subtracted from all other data, resulting in a reduced invariant Q*, according to where ∆ρ 2 = |ρ ZnO − ρ MeOH | 2 . In Figure 6, the obtained cluster volume fraction φ ZnO-cluster,inv (t) ( In Figure 1a, the recorded slit-smeared SAXS intensities I(q) were plotted as a functi of the scattering vector q using a double-logarithmic representation. The scattering curve PVP in pure methanol ϕPVP = 7.1 × 10 −2 (8.6 mM) ( ) shows a Guinier region at low with almost constant scattering intensity. As q increases, a power law decay is observed un a minimum is reached at q ≈ 1 Å -1 , followed by the intermolecular C-C chain interaction pe [66] at q ≈ 1.5 Å −1 . Similar behavior was found when 12.5 mM ZnAc2 ( ) or 26 mM TEAOH ( )were added while keeping the polymer amount constant at ϕ = 7.1 × 10 −2 . However, for the final composition of the precursor solution ( instead of a Guinier region for q < 0.1 Å −1 , a power law decay with I(q) ≈ q −1.4 can be see Thus, the presence of both ZnAc2 and TEAOH results in a significantly different structu of the polymer.
(where both ZnAc2 and TEAOH are present) in more detail, the power law decay (I ≈ q −1.4 ) recorded in the low q limit of the in-house SAXS instrument provides little info mation on the global structure and size of the polymer. This information can be obtained extending the q-range towards much smaller values using the USAXS/SAXS setup at the ID-C beamline at the Advanced Photon Source at the Argonne National Laboratory [58,5 The inset in Figure 1 shows a good agreement between the desmeared in-house ( ) and USAXS/SAXS ( ) data. In addition, the combination of the da clearly shows the transition of the power law into a Guinier region at q < 0.01 Å −1 . Utilizi Guinier's approximation for low q (ln I(q) ≈ ln I0 − (Rg 2 /3) q 2 ), we obtained Rg(PVP) = 6.3 0.3 nm, which corresponds to a fourfold increase compared to Rg(PVP) in pure methan and in solutions containing only ZnAc2 or TEAOH. This significant change in the PV structure caused by the increased ion concentration due to the presence of both ZnO pr cursor and base is most likely related to a stronger repulsion of the segments within t polymer chain, leading to a further increase in the chain stiffness. This hypothesis is su ported by the increase in mesh size determined from the analysis at medium q's to ξPV 2.9 ± 0.2 nm. Moreover, ZnAc2 might start to hydrolyze even at 25 °C, changing its cha flexibility so that the first particle nuclei could have formed within the PVP. However, th reaction is significantly accelerated only by increasing the temperature. The FW model assumes continuous nucleation of ZnO particles. This can be inde observed in the volume fraction profile of the ZnO particles ( ) for T = 58 °C in Figure where an initial increase is followed by reaching an almost constant value, while the vo ume fraction of the cluster ( ) follows a sigmoidal profile. These results suggest that ne ZnO particles are formed continuously throughout the reaction, feeding the cluste which in turn grow simultaneously. Hence, at intermediate times, an equilibrium is esta lished between ZnO particles' formation and consumption. where ∆ 2 = |ρZnO − ρMeOH | 2 . In Figure 6, the obtained cluster volume fraction ϕZnO-cluster,inv(t) ) for T = 58 °C is plotted as a function of time. Comparing the cluster volume fractions obtained from the model analysis of the USAXS data and the invariant, i.e., ϕZ cluster,I(q) and ϕZnO-cluster,inv, slightly lower values are found for the latter, which might be consequence of the finite q-range used for the invariant determination as well as the ap proximation of spherical clusters with compact packing of ZnO particles and a homog neous scattering length density distribution.
) for T = 58 • C is plotted as a function of time. Comparing the cluster volume fractions obtained from the model analysis of the USAXS data and the invariant, i.e., φ ZnO-cluster,I(q) and φ ZnO-cluster,inv , slightly lower values are found for the latter, which might be a consequence of the finite q-range used for the invariant determination as well as the approximation of spherical clusters with compact packing of ZnO particles and a homogeneous scattering length density distribution.
The profiles of the cluster volume fraction φ ZnO-cluster,inv (t) were also described using the FW model (in Figure S5 of the Supplementary Materials, the profiles obtained for T = 58 • C and T = 63 • C are shown for comparison). A comparison of the parameters ( Table 2) obtained from the analysis of the R 3 (t)-data and φ ZnO-cluster,inv (t)-data, respectively, reveal the same growth rate constants k 2 and induction times t ind , which are similar within the measurement uncertainty. The smaller value of the nucleation rate constant for the φ ZnO-cluster,inv (t)-data might also be a consequence of the finite q-range used for the invariant determination. Consequently, larger S-values in the analysis of the φ ZnO-cluster,inv (t)-data are obtained.
To conclude the in situ study of ZnO particle formation and growth, we found that shortly after heating the precursor solution, ZnO nanoparticles of the order of R ZnO ≈ 1.5 nm are initially formed, followed by the formation of clusters. An increase in the reaction temperature from 58 • C to 63 • C resulted in the acceleration of the reaction and the formation of larger clusters. From the time evolution of the cube of radius, a linear relationship is apparent at intermediate times. Depletion forces caused by the high volume fraction of PVP might explain the relatively high growth rate. A quantitative description of the time evolution of the ZnO nanoparticle cluster radius, the corresponding cube of the radius, as well as the volume fraction, could be achieved by the Finke-Watzky model [62][63][64][65] with slow, continuous nucleation of individual ZnO particles followed by autocatalytic growth of the clusters, as illustrated in Figure 7. From the analysis of the kinetic data, the higher ratio of growth to nucleation rate explains that slightly larger clusters are formed at the higher temperature of 63 • C. formation of larger clusters. From the time evolution of the cube of radius, a linear relationship is apparent at intermediate times. Depletion forces caused by the high volume fraction of PVP might explain the relatively high growth rate. A quantitative description of the time evolution of the ZnO nanoparticle cluster radius, the corresponding cube of the radius, as well as the volume fraction, could be achieved by the Finke-Watzky model [62][63][64][65] with slow, continuous nucleation of individual ZnO particles followed by autocatalytic growth of the clusters, as illustrated in Figure 7. From the analysis of the kinetic data, the higher ratio of growth to nucleation rate explains that slightly larger clusters are formed at the higher temperature of 63 °C.

Conclusions
A crucial step for the synthesis of metal oxide inverse opals is the mineralization of the voids of a close-packed polymer nanoparticle templates via the chemical bath deposition (CBD) process. Intrigued by the vision of tuning the size and shape of ZnO particles, we aimed to gain more insight into the particle formation in an additive containing precursor solution by in situ time-resolved ultra-small-and small-angle X-ray scattering (USAXS/SAXS).
First, we used in-house SAXS and USAXS/SAXS to study the structure of polyvinylpyrrolidone (PVP) applied as structure directing agent for the ZnO formation in the methanolic precursor solution at T = 25 °C. We found that at the volume fraction of ϕPVP = 7.1 × 10 −2 used in the CBD process, the polymer forms a network of entangled chains in which ions of the ZnO precursor salt and the organic base are distributed, causing the polymer chains to adopt a semiflexible behavior.
We then followed the ZnO formation using in situ USAXS/SAXS by heating the precursors solution from T = 25 °C to T = 58 °C and 63 °C, respectively. A significant increase in the intensity of the forward scattering and the formation of an additional weak shoulder was observed after 13 min, both of which occur more rapidly at T = 63 °C. The origin of the two scattering contributions was clarified from post-synthesis TEM images, which clearly showed the presence of two ZnO species: ZnO nanoparticles with a radius of RZnO,TEM = 1.5 ± 0.5 nm and clusters of these with RZnO-cluster,TEM = 8.5 ± 1.4 nm and RZnOcluster,TEM = 9.7 ± 1.9 nm at T = 58 °C and 63 °C, respectively. Through this additional information, we were able to quantitatively analyze the time-resolved USAXS/SAXS data by additively combining the scattering contribution of the PVP network with the scattering

Conclusions
A crucial step for the synthesis of metal oxide inverse opals is the mineralization of the voids of a close-packed polymer nanoparticle templates via the chemical bath deposition (CBD) process. Intrigued by the vision of tuning the size and shape of ZnO particles, we aimed to gain more insight into the particle formation in an additive containing precursor solution by in situ time-resolved ultra-small-and small-angle X-ray scattering (USAXS/SAXS).
First, we used in-house SAXS and USAXS/SAXS to study the structure of polyvinylpyrrolidone (PVP) applied as structure directing agent for the ZnO formation in the methanolic precursor solution at T = 25 • C. We found that at the volume fraction of φ PVP = 7.1 × 10 −2 used in the CBD process, the polymer forms a network of entangled chains in which ions of the ZnO precursor salt and the organic base are distributed, causing the polymer chains to adopt a semiflexible behavior.
We then followed the ZnO formation using in situ USAXS/SAXS by heating the precursors solution from T = 25 • C to T = 58 • C and 63 • C, respectively. A significant increase in the intensity of the forward scattering and the formation of an additional weak shoulder was observed after 13 min, both of which occur more rapidly at T = 63 • C. The origin of the two scattering contributions was clarified from post-synthesis TEM images, which clearly showed the presence of two ZnO species: ZnO nanoparticles with a radius of R ZnO,TEM = 1.5 ± 0.5 nm and clusters of these with R ZnO-cluster,TEM = 8.5 ± 1.4 nm and R ZnO-cluster,TEM = 9.7 ± 1.9 nm at T = 58 • C and 63 • C, respectively. Through this additional information, we were able to quantitatively analyze the time-resolved USAXS/SAXS data by additively combining the scattering contribution of the PVP network with the scattering contributions of the individual ZnO particles and their clusters. This analysis revealed that the radius of the formed individual ZnO particles (R ZnO,f = 1.5 ± 0.3 nm) remains constant over time for both temperatures, while the radius of the particle clusters increases from the induction time onwards to R ZnO-cluster,f = 10.5 ± 1.9 nm at T = 58 • C and, R ZnO-cluster,f = 11.8 ± 2.1 nm at T = 63 • C, with R 3 (t) following a sigmoidal profile.
From the linear relationship of R 3 (t) at intermediate times, relatively large rate cluster growth constants were determined, which might indicate PVP-induced depletion forces. Finally, using the Finke-Watzky two-step model, we were able to quantitatively describe the ZnO formation with slow, continuous nucleation of individual ZnO particles followed by autocatalytic growth of the clusters. A faster growth rate of the clusters compared to the nucleation of new ZnO particles was observed at higher temperatures, which thus explains the formation of larger clusters at 63 • C.
The determination of the kinetics of ZnO formation, synthesized by CBD using a methanolic solution and PVP as a stabilizing and structuring agent, will provide us with valuable insights into the understanding of ZnO deposition in the voids of densely packed polymer nanoparticle templates, where both surface and confinement effects are expected.
Supplementary Materials: The following supporting information can be downloaded at https: //www.mdpi.com/article/10.3390/nano13152180/s1: SAXS fitting parameters of PVP in methanol at φ PVP = 7.1 × 10 −2 before and after adding 12.5 mM ZnAc 2 and 26.3 mM TEAOH obtained with the polymer with excluded volume model (Table S1) or the correlation length model (Table S2) combined with a Gaussian, respectively; Figure S1: Slit-smeared SAXS curves of PVP in methanol at various volume fractions φ; SAXS fitting parameters from PVP in methanol at φ PVP = 1.0, 2.0, 4.0, and 7.1 × 10 −2 obtained with the polymer with excluded volume model (Table S3) and the correlation length model (Table S4) combined with a Gaussian, respectively. Figure S2: Obtained fitting parameters for PVP in methanol at various volume fractions φ PVP ; Figure S3: Final fit for the scattering curve of the precursor solution after heating to T = 58 • C for t = 87 min with the respective scattering contributions. Figure S4: TEM image taken after the ZnO synthesis for T = 63 • C with the corresponding volume-weighted size distribution of the ZnO particles and clusters, described by a Gaussian size distribution, respectively. Figure S5: Time evolution of the volume fraction φ ZnO-cluster,inv of ZnO particle clusters determined from the invariant analysis (φ invariant ) of the USAXS curves up to q ≤ 0.04 Å −1 . The profiles were described by the two-step Finke-Watzky model (solid lines) from which the induction time t ind (stars) were calculated.